Is it an $NP$-hard problem?

You're given an undirected graph $G(V,E)$ with vertex weight $w: V \to \mathbb{N}$ and a function $\mathrm{max}$-$\mathrm{visit}: V \to \mathbb{N}$ and a number $W$.

Does there exists a path in $G$ with total weight $W$ that does not visit any vertex $v$ more than $\mathrm{max}$-$\mathrm{visit}(v)$ times?


So this is a simple solution to this question. Given an instance of Hamiltonian path problem $G(V, E)$. Just set $w$ and $\mathrm{max-visit}$ to be equal to $1$ everywhere and $W=|V|$. A path in $G$ satisfies the requirements in our problem now corresponds to a Hamiltonian path in the original instance, and vice versa.

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